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New Avenues for History in Mathematics Education: Mathematical Competencies and Anchoring

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Abstract

The paper addresses the apparent lack of impact of ‘history in mathematics education’ in mathematics education research in general, and proposes new avenues for research. We identify two general scenarios of integrating history in mathematics education that each gives rise to different problems. The first scenario occurs when history is used as a ‘tool’ for the learning and teaching of mathematics, the second when history of mathematics as a ‘goal’ is pursued as an integral part of mathematics education. We introduce a multiple-perspective approach to history, and suggest that research on history in mathematics education follows one of two different avenues in dealing with these scenarios. The first is to focus on students’ development of mathematical competencies when history is used a tool for the learning of curriculum-dictated mathematical in-issues. A framework for this is described. Secondly, when using history as a goal it is argued that an anchoring of the meta-issues in the related in-issues is essential, and a framework for this is given. Both frameworks are illustrated through empirical examples.

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Notes

  1. Examples are: The International Study Group on the Relations between the History and Pedagogy of Mathematics (HPM), which together with the International Group for the Psychology of Mathematics Education (PME), became one of the first affiliated study groups of the International Commission on Mathematical Instruction (ICMI) in 1976 (Fasanelli and Fauvel 2007). HPM holds a satellite conference to ICME (which itself also has a topic study groups on history) every fourth year, alternating with the French initiative, the now quadrennial European Summer University on the History and Epistemology in Mathematics Education (ESU), which began in 1993 (Barbin et al. 2007). The years in between HPM and ESU, the Congress of the European Society for Research in Mathematics Education (CERME) has a working group dedicated to History in Mathematics Education, which began in Lyon 2009 and takes place again at CERME 7 in Poland, 2011. On top of this the HPM Americas section holds meetings every year. All these conferences and meetings produce proceedings thus adding to the amount of papers on history in mathematics education. The increasing interest can also be seen from papers published in more general journals on mathematics education such as Educational Studies in Mathematics (ESM) and For the Learning of Mathematics (FLM) of which special issues on history in mathematics education have appeared in 2007 and 1991, respectively. And also entire books on the subject are available: Fauvel (1990), Swetz et al. (1995), Jahnke et al. (1996), Calinger (1996), Katz (2000), Shell-Gellasch and Jardine (2005), and most importantly Fauvel and van Maanen (2000)—the most comprehensive sample on the topic to date.

  2. For examples see Fauvel and van Maanen (2000).

  3. For a survey of empirical studies on the use of history in mathematics, see Jankvist (2009c).

  4. See Jankvist (2009a) for an elaboration of this.

  5. The same argument can be put forward against a use of history as a motivating tool, plenty of other topics, perspectives, etc. may motivate students to study and learn mathematics.

  6. This refers to a reading of the past in which one tries to find the present. See also for example Schubring (2008), Kjeldsen (2010a), or Kjeldsen and Blomhøj (Accepted).

  7. The construct of anchoring has also been addressed and discussed as part of a plenary talk at the 6th ESU in Vienna. Interested readers are thus referred to the forthcoming proceedings from this conference.

  8. KOM is a Danish abbreviation for competencies and mathematics learning.

  9. See footnote 19.

  10. Recent history of mathematics has taken similar approaches, see e.g. Epple (2000), Corry (2004), Kjeldsen et al. (2004), Kjeldsen (2009a).

  11. See also Epple (2004, p. 133).

  12. The term ‘multiple perspective’ approach to history is borrowed from the Danish historian Bernard E. Jensen (Jensen 2003, p. 16). For its use in history in mathematics education, see also Kjeldsen (2010a), Kjeldsen (forthcoming a), Kjeldsen and Blomhøj (Accepted).

  13. These concepts have been introduced into the historiography of mathematics by Epple (2004) inspired by a methodological framework originally developed by Rheinberger (1997) for the study of experimental science.

  14. The project work for the 2 year science introductory program and its educational purposes are described in Blomhøj and Kjeldsen (2009). The integration of history and philosophy of mathematics within the 2 year science program is described and discussed in Kjeldsen and Blomhøj (2009).

  15. A discussion of the first student project (Sect. 6.1) with respect to developing mathematical competence and turning meta-discursive rules into objects of reflections may be found in Kjeldsen (forthcoming a) and Kjeldsen and Blomhøj (Accepted), respectively.

  16. All quotes from the projects have been translated from Danish into English by us.

  17. The students’ formulation of their problem for their project work is always a source of critique and discussions that in most cases go on throughout the entire semester. In this case, e.g. the students’ formulation of the second question can be interpreted as though they have a platonic view of mathematics.

  18. Theorem 23 of Archimedes’ Quadrature of the Parabola states that: “given a series of areas A, B, C, D, … Z of which A is the greatest, and each is equal to four times the next in order, then A + B + C + ··· +  Z + 1/3 Z = 4/3 A”.

  19. A framework for how such an approach can be implemented in secondary mathematics education on a small scale was presented at the workshop Does history have a significant role to play for the learning of mathematics? Multiple perspective approach to history, and the learning of meta level rules of mathematical discourse at the ESU 6 in Vienna, July 19–23, 2010. Interested readers are thus referred to the forthcoming proceedings from this conference.

  20. If including epistemic objects and techniques as a general topic and issue in itself, see Jankvist (2009c) and Jankvist (2010a).

  21. For an example of relationships between research in pure and applied mathematics of the 20th century, see Kjeldsen (2010b).

  22. Yet another example of a general topic and issue (also related to the third type of overview and judgment) could be that of the old discussion of mathematics being discovered or invented (e.g. Hersh 1997). See Jankvist (2009c) for a more thorough discussion of the above mentioned general topics and issues.

  23. The fixed curriculum is tested on a national written exam and the remaining 1/3 is tested on a local oral exam (together with parts of the fixed curriculum).

  24. The two materials, Jankvist (2008b) and Jankvist (2008c), are available online as IMFUFA Texts no. 459 and 460, respectively. See http://milne.ruc.dk/ImfufaTekster/

  25. The students’ insights and possible changes in beliefs and views were evaluated by giving them questionnaires and conducting individual interviews before, in between, and after the implementations of the two modules. For an elaboration of this, see Jankvist (2009c).

  26. In the present paper we have tried to illustrate how to establish if students posses and/or develop any of the eight mathematical competencies or the type of overview and judgment regarding history. However, the question of how to judge the degree of possession of a competency by someone is a different matter. In the KOM-report the authors distinguish between three dimensions that can be used to evaluate to what degree a person possesses a given competency: coverage, span of action, and technical level. Due to the KOM-report still being fairly new and because its aspects are still in the process of finding their way into actual teaching and mathematics programs, research on these aspects are only just emerging, and we will not go into further details on this issue here.

  27. Phrased by Victor Katz at the HPM Americas’ annual meeting in Washington DC, 2010. See minutes: http://www.hpm-americas.org/wp-content/uploads/2010/04/Minutes.pdf

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  1. Department of Mathematics and Computer Science, Southern University of Denmark, Odense, Denmark

    Uffe Thomas Jankvist

  2. IMFUFA, Department of Science, Systems and Models, Roskilde University, Roskilde, Denmark

    Tinne Hoff Kjeldsen

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  1. Uffe Thomas Jankvist
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Jankvist, U.T., Kjeldsen, T.H. New Avenues for History in Mathematics Education: Mathematical Competencies and Anchoring. Sci & Educ 20, 831–862 (2011). https://doi.org/10.1007/s11191-010-9315-2

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